Set and Metric Space: Open set, Dense Set, Countability, Compact and connected sets, Monotonic class of sets, Additive class of sets, Metric space, continuous functions of metric spaces, Application of metric spaces in statistics.

Sequences: Introduction to sequences, monotonic sequence, bounded sequence, convergence sequence, properties of sequence, Cauchy’s sequence.

Series: Introduction, Convergence principle, Convergence and absolute convergence of series, Cauchy’s convergence, Comparison test,  Ratio test, Root test, Integral test, Abel’s Lemma, Dirichlet’s test, Abel’s test for conditional convergent power series, Rearrangement of absolute convergent series, Multiplication of absolutely convergent series.

Vector Valued functions of several variables: Introduction, Linear transformation and matrices, Continuity and differentiability of transformations, The inverse function theorem, The implicit function theorem.

Vectorization and Matrix Differentiation: Definition, Function of matrices, Differentiation of different types of matrix functions with respect to vector and matrix, Differentiation of quadratic form.

Riemann Integral: Introduction, Geometrical interpretation, The existence of the Riemann integral of a continuous function, Simple properties, First and second mean value theorem, Convergence and absolutely convergence of improper and infinite integrals, Term by term integration and differentiation.

Multiple Integral: Double integral, Triple integral, Line and surface integral.

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